We present a general framework for designing efficient algorithms for unsupervised learning problems, such as mixtures of Gaussians and subspace clustering. Our framework is based on a meta algorithm that learns arithmetic circuits in the presence of noise, using lower bounds. This builds upon the recent work of Garg, Kayal and Saha (FOCS 20), who designed such a framework for learning arithmetic circuits without any noise. A key ingredient of our meta algorithm is an efficient algorithm for a novel problem called Robust Vector Space Decomposition. We show that our meta algorithm works well when certain matrices have sufficiently large smallest non-zero singular values. We conjecture that this condition holds for smoothed instances of our problems, and thus our framework would yield efficient algorithms for these problems in the smoothed setting.

翻译：我们提出一个通用框架，用于设计高斯混合模型和子空间聚类等无监督学习问题的高效算法。该框架基于一个元算法，该算法利用下界技术在含噪环境下学习算术电路。这一工作建立在Garg、Kayal与Saha (FOCS 20)近期提出的无噪算术电路学习框架之上。元算法的核心是为一个名为鲁棒向量空间分解的新问题设计的高效算法。研究表明，当某些矩阵具有足够大的最小非零奇异值时，该元算法表现良好。我们推测该条件适用于所研究问题的光滑实例，因此该框架可在光滑环境下为这些问题提供高效算法。